International Journal of Computational Engineering Research (IJCER)
Definite Integral Relation Involving H-Function of Multivariable Function and General Class of Polynomial
Jyoti Shaktawat
1,Ashok Singh Shekhawat
21Research scholar, Suresh Gyan Vihar University, Jaipur, Rajasthan (India)
2Department of Mathematics, Suresh Gyan Vihar University, Jaipur, Rajasthan (India) Corresponding Author: * Jyoti Shaktawat
--- --- Date of Submission: 27-12-2017 Date of acceptance: 09-01-2018 --- ---
I. INTRODUCTION
The H-function of several complex variable is defined by Srivastava and Panda [14] as:
0, :(u',v');...; (u(r)(r),v(r)(r)) [(p): ' ;...;(r)(r)]:[(q'): '];...;[(q(r)(r)):(r)(r)];
1 r A,C:[B',D'];...;[B ,D ] (s): ';...; ]:[( t'): '];...;[(t ): ]; 1 r
H[x ,..., x ] H
x ,..., x
…(1.1)The Fox’s H-function [2]:
Gg G L G
1 g 0 G MP
mP NQ (nQ R L,
Q
P,
x
N
! G x 1
H
…(1.2)where
G j j P
1 R G j j j Q
1 L j
G j j R
1 G j j j L
g j
1 j
G
M m N
n 1
M m 1 N
n
and
g g G
G)
The H-function of multivariable in (1.1) converges absolutely if
ii
T
2 (x 1
arg
…(1.3)where
ABSTRACT
In this paper, we obtain an integral relation containing to a product of Fox’s H-function [2], , general class of multivariable polynomials srivastava and Garg [11],generalized polynomials srivastava [10]
and H-function of several complex variables given by srivastava and panda [14] with general argument of quadratic nature. This paper is capable of yielding numerous result involving classical orthogonal polynomials.
Key Words: Fox’s H-function, general polynomials, general class of polynomials, generalized lauricella function, G-function, multivariables H-function
0
T
ij D(i)
u(i) 1 j (i)
j u(i)
1 j (i) j C
1 j (i)
j B(i)
v(i) 1 j (i)
j v(i)
1 j (i)
j A
1 i j
i (1,…,r) …(1.4)
general class of polynomials introduced by srivastava [10] is as follows
Z
M
k N, Mk (N/M)
0
N
B
! N) Z) (
(
S
…(1.5)
Where M,N are arbitrary positive integer and the coefficients BN,K are arbitrary constant ,real or complex.
Srivastava has defined and introduced the general polynomials ([10], p.185, eq.(7))
s s Ms s
1 1 M1 s 1
s M N
s 0 M1
N1
1 0 r
1 Ms M1
Ns N1
N N
z z S
ss 1 1 s s 1
1
N z z
[N
B
…(1.6)where Ni = 0,1,2,…, i = (1,…,s); M1 ,…, Ms are arbitrary positive integers and the coefficients B [N1,1 ;…;
Ns,s ] are arbitrary constants, real or complex.
II. THE MAIN INTEGRAL RESULT Here we obtain following integral :
w
( a bw cw ) H a bw w cw ( ( m n , , N M ) )
Q QP P 2
R L,
Q P, 2 / 3 2 1
0
s 1
Ms , . . . , M1
Ns , . . . , N1
n
s 2 n
1 2
cw bw a z w ,..., cw
bw a z w S
.
n
2 M
cw bw a z w S .
Ncw dw bw a x w ,..., cw
bw a x w H .
r 1
r 2 1 2
)
! ( ) N ... (
! ) N ( F G!
) 1 B (
! N) ... (
c
s GM s
1 M 1
g G k N, Mk (N/M)
0 ] M / N [
0 ] M / N [
0 L
1 g 0 G
s s 1
1 s
s
s 1 1
1
s
1
...( z )
z ] N
;...;
,
B[N
1
1 s
s 1 s z
i i 1s 1 i G
) ca 2 b (
, ]
;...;
' : s) [(
],
;...;
: n [
) ca 2 ( x
) ca 2 b ( x H
.
r) (
r 1 i i s
1 i G
r 1
r 1 (r))
v (r), (u
;. . . ; ) v' , u' ( : 1 , 0
(r)] D (r), [B
;. . . ; ] D' , [B' : 1 C 1, A
b
] : ) [(t
;...;
] ' : ) t' [(
: ]
;...;
2 : n 1 [
] : ) [(q
;...;
] ' : ) q' [(
: ]
;...;
' : p) ( [
r) ( (r) r
1 i
i s
1 i
r) ( (r) r)
(
…(2.1)
provided that Re(a) > 0, Re(b) > 0, c > 0 and
) (i' )
(i' j'
) (i' ' j'
i r
1 j i'
j
2 j 1,..., M and j' 1,..., u
t Re N min
n Re min
Proof:
For getting the result (2.1) first we express the Fox H-function and a general polynomials in form of series and the H-function of multivariable in terms of Mellin-Barnes contour integrals. Now interchanging the order of summations and integration which is permissible under the stated condition, we get
)
! ( ) N ... (
! ) N ( ) 1 B (
! N)
... (
Gs M s
1 M 1 G k N, Mk (N/M)
0 ] M / N [
0 ] M / N [
0 L
1 g 0 G
s s 1
1 s
s
s 1 1
1
.
B[N
1,
1;...; N
s
s] ( z
1)
1...( z
s)
s z
r r 1 1 r r 1 1 r 1 Ir
I1
r
x x
i) 2
1
1 [ ni i 11 ... r r) s1 i G
0
w .
nl
. ( a bw cw
2)
2d d
1... d
r,
) 3 r . . . r 1 nl 1 i- ni s
1 Gi
(
w
…(2.2)On solving above w-integral by help of known theorem (Saxena [8]) and reinterpreting the result obtained in terms of H-function of r variable, we get the desired result.
III. PARTICULAR CASES
(a) When we put = A = C = 0 in (2.1), we get the following integral result
w
( a bw cw ) H a bw w cw ( ( n m , , N M ) )
Q QP P 2
R L,
Q P, 2 / 3 2 1
0
1 s Ms
, . .. , M1
Ns , . .. , N1
n
s 2 n
1 2
cw bw a z w ,..., ) cw bw a z w S
.
.
n
N 2
cw bw a z w S .
M] dw : d [
] : ) b [(
cw bw a x w H
.
(i) (i)i) ( (i) i 2
r
1 i
(i) 1 v (i), u
D(i) (i),
B
) ) (
N ... ( ) N ) (
1 B ( N)
... (
1 M s M,G Mk
(N/M) ] M / N ] [ M / N L [
s s 1
1 s
s 1
1
(r)) v (r), (u );. . . ; v' , u' ( : 1 , 0
(r)] D (r), [B ];. . . ; D' , [B' : 1, 1, i i s
1 i G
1 s
H )
ca 2 b (
) z ...(
z ] , N
;...;
, [N B ( .
1 n
s 1 s s 1 1
z
l
] : ) [(t
;...;
] ' : ) t' [(
: ]
;...;
2 : n 1 [
] : ) [(q
;...;
] ' : ) q' [(
: ]
;...;
: n [
) ca 2 b ( x
) ca 2 b ( x
r) ( (r) r
1 i
i s
1 i G
r) ( (r) r
1 i i s
1 i G
r 1
r 1
…(3.1) valid under the same condition which is obtained from (2.1).
(b) If = A, u(i) = 1, v(i) = B(i) and D(i) = D(i) + 1, i (1,…,r) the result in (2.1) reduces to the following integral transformation:
w
( a bw cw ) H a bw w cw ( ( n m , , N M ) )
Q QP P 2
R L,
Q P, 2 / 3 2 1
0
s 1
Ms , . . . , M1
Ns , . . . , N1
n
s 2 n
1 2
cw bw a z w ,..., cw
bw a z w S
.
n N 2
cw bw a z w S .
M
r)] ';. . . ;( : p) ( 1 [
r)] ';. . . ;( : s) ( 1 [ s (r) 1
B B';. . . ; : A
D(r)
;. . . ; D' :
C 1 2 r 2
cw bw a x w ,..., cw
bw a x w E
.
]
dw
r) :( (r)) (b [1
;. .. ; ] ' : ) q' ( 1 [ :
r)] :( (r)) (d [1
;. .. ; ] ' : ) t' ( 1
[
)
! ( ) N ... (
! ) N ( F G!
) 1 B (
! N) ... (
c
s GM s
1 M 1
g G k N, Mk (N/M)
0 ] M / N [
0 ] M / N [
0 L
1 g 0 G
s s 1
1 s
s
s 1 1
1
.
B(N
1,
1;...; N
s
s)
1 s ni i 1s
1 i G s
1
...( z ) ( b 2 ca )
z .
z
A 1: B';...; B(r) C 1 : D';...; D(r)
s
G i i i
i 1 s
G i i i
i 1
(1 n ( k )
3 E
( n ( k )
2
],
;...;
' : s) ( 1 [
],
;...;
: n 1
) [ ca 2 b ( x ,..., ) ca 2 b ( x .
(r)
r 1 i i s
1 i G r
1
r 1
] );
t ( [1
;...;
] ' : ) t' ( 1 [ : ]
;...;
: 2 n
[ 3
] : ) q ( 1 [
;...;
] ' : ) q' ( 1 [ : ]
;...;
' : p) ( 1 [
r) ( (r) r
1 i i s
1 i G
r) ( (r) r)
(
…(3.2)
provided that Re(a) > 0, Re(b) > 0,c > 0, the series on the right side exists.
(c) If ',…,(r) = ' ,…, (r) = ' ,…, (r) = ' ,…, (r) = 1,…,r = ' ,…, (r) in (2.1), we obtain the following integral result:
w
( a bw cw ) H a bw w cw ( ( n m , , N M ) )
Q QP P 2
R L,
Q P, 2 / 3 2 1
0
1 s Ms
, . .. , M1
Ns , . .. , N1
n
s 2 n
1 2
cw bw a z w ,..., ) cw bw a z w S
.
n
2 M
N
a bw cw
z w S .
dw
) );...;
( : ) (
) );...;
( : ) 2 ( 1/
2 r '
1/
1 ) v , u
;...;
) v' , u' ( : 0,
] D , [B
;...;
] D' , [B' : C A,
(q(r) q' p
(t(r) t' s r
(r) ( (r)
(r)
(r)
,
cw bw a x w ,..., cw bw a x w F
.
)
! ( ) N ... (
! ) N ( F G!
) 1 B (
! N) ... (
c
s G, M s
1 M 1
g G k N, Mk (N/M)
0 ] M / N [
0 ] M / N [
0 L
1 g 0 G
s s 1
1 s
s
s 1 1
1
.
B(N
1,
1;...; N
s
s)
1 s ni i 1s 1 i G s
1
...( z ) ( b 2 ca )
z .
z
B(r) B';. . . ; : 1 A
D(r) D';. . . ; : 1 C
E 2 n
( 3
n 1
(
i i s
1 i G
i i i s
1 i
G
1 1/
r 1 '
1/
1 ) v , (u
;...;
) v' , u' ( : 1 0,
D , [B
;...;
] D' , [B' : 1 C 1,
A
x ( b 2 ca ) ,..., x ( b 2 ca ) F
.
(r)(r) (r)(r) r
(r) i
i s
1 i G
(r) i
i s
1 i G
q );...;
(q' : p) ( , n
) (t );...;
t' ( 2 : n 1 s),
(
…(3.3)
provided that Re (a) > 0, Re (b) > 0, c > 0; (i) > 0 (i = 1,…,r), 2(u(i) + v(i)) > (A+C+B(i)+ D(i))
Re (n N min Re(t 2
min
2 and D 2
B 2 C 2 v A u (z
arg
(i) (i) j
u j 1 r
1 j i
M j j 1
(i) (i)
(i) (i) i
IV. CONCLUSION
The result so obtained may be found useful in several interesting situation appearing in the literature on Mathematical analysis applied Mathematics and Mathematical physics. These results are basic in nature and likely to find useful application in the study of simple and multivariable hypergeometric series.
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